[math-06] Verify equilibrium and plasma-background formulas

[marjohma]

Summary

• derive multispecies equilibrium, collision, thermodynamic-force, and geometry formulas
• correct shared constants and background formula inconsistencies across KIM and KiLCA
• add deterministic KIM and KiLCA oracle consumers

Why

Both solvers use source-checked equilibrium quantities with consistent units and species sums.

Validation

Validated cumulatively at the stack head:

• cmake --build build -j2
• ctest --test-dir build -E 'test_rhs_balance|test_periodic_convergence|test_periodic_vs_global' --output-on-failure — 35/35 passed

Known red gates are intentionally excluded from that command: the pre-existing test_rhs_balance baseline failure, the periodization-deformation acceptance gate, and the unresolved periodic-vs-global 5% agreement gate.

Stack

Base: fix/kim-fp-188-matched-bvp
Head: fix/kim-fp-198-equilibrium-oracle

Closes #198

[marjohma]

Review (standards + spec vs #198) — keeping this in draft. Two distinct things are bundled here, and they deserve different fates.

1. Undisclosed production-numerics changes (the main reason for the hold). Four real fixes hide inside a fix(background): verify… commit:

• species_mod.f90 ion-ion Coulomb logarithm rewritten: the old Lii lacked the NRL sqrt and had misplaced parentheses — at n=2e13 cm⁻³, T=800 eV this shifts Lii ~5.7 → ~17.4, i.e. ~ν_ii×3 in every run, single-ion included; the ion-partner loop also went upper-triangle → full (multi-ion change).
• constants_mod.f90:8: sol literal lacked _dp and was stored single-precision (29979246592 vs 29979245800) — every c-dependent result shifts ~2.6e-8. Correct fix, disclosed nowhere.
• compute_rg_cell_centers: derived quantities now recomputed from averaged primitives (was: averaged derived values), plus the x2_cc omega_c(j) → omega_c_cc(j) index fix.
• Ion-density fallback n_e·Z_s/ΣZ → n_e/ΣZ (only the new form satisfies ΣZ·n=n_e), with new hard error stops.

These all look physically right — and the new KIM oracle test genuinely exercises production species_m code at 5e-12 and would fail against the old formulas, which is exactly what an oracle should do. But: the golden-record CI (rtol 1e-8) will flag the Lii and sol changes, so a baseline re-bless is needed, and result-changing physics fixes shouldn't ride silently in a verification commit. Suggest splitting them into an explicit fix(KIM): commit/PR with the numerics impact stated, then rebasing the verification on top. (The KiLCA calc_back.cpp extraction, in contrast, was verified behavior-preserving term-by-term — that part is clean plumbing.)

2. Verification-layer gaps (the recurring family pattern).

• "Prove sum_s Z_s n_s = n_e for arbitrary mixtures": no symbolic proof — the WL check sets nd = ne − 6·nCarbon and then verifies nd + 6·nCarbon == ne (tautology); the Fortran side is fixed D/C examples plus a runtime residual.
• Production calculate_equil and KiLCA calculate_equilibrium/find_f0_parameters are never executed against the oracle — the geometry/current/force-balance/E_r rows (kinds 1/5/6) are only self-consistency-checked in the Fortran test, and the KiLCA test skips kind≠4. FORMULA_INDEX's "independent [math-06] Verify equilibrium and plasma-background formulas #198 fixture" for KIM-03/KILCA-05/06 overstates this; the WL formulas there are transcriptions.
• The issue's own success-criteria regex test_(periodic_background|profile_input|plasma) does not match the new test_kilca_plasma_background_oracle — the KiLCA gate never runs under the spec's command.
• The four negative fixtures mutate the WL script's own constants/toys, not production formulas; several checks ("zero flow removes rotation term" compares a call to an identical call) cannot fail unmutated.
• Untested production paths: rescale_density=.true. branch, A1_cc/A2_cc.
• Hardcoded counts again (17 rows, kind counts, 29 sites) — mitigated by the row-wise 1e-30 regeneration cross-check.

Genuine wins worth keeping as-is: derivatives-after-periodization is properly verified against production (test_periodic_background_build.f90:292-315), both oracle CTests are registered and read the committed .dat, provenance is good, and the q-inversion/Lorentz-force symbolic checks are real.

Minor: two >100-col lines in species_mod.f90 and two in the new .wl.